Discuss the validity of the following statement about linear systems. If the num

Question

Discuss the validity of the following statement about linear systems. If the number of leftmost ones is equal to the number of variables and the system is consistent, then the system has exactly one solution. Choose the correct choice below. A. The statement is always true, because if the number of leftmost ones is equal to the number of variables, each leftmost variable in a reduced system appears in one and only one equation and no equation contains more than one leftmost variable. The statement is false. A counterexample is [1 0 0 0 1 0 | 1 1 1]. C. The statement is always true, because if the number of leftmost ones is equal to the number of variables and the system is consistent which means there are no contradictions, the equations for the reduced matrix are in the form x_1 = a_1, x_2 = a_2, ..., x_i a_i where a_1.a2, ... a_i are real numbers. D. The statement is false. A counterexample is [1 0 0 0 1 0 | 1 1 0].

Explanation / Answer

C) option is correct as if the equation has same number of variables as that of leftmost ones and is consistent .

Then two possibilitues are there either unique solution or infinitely many solutions . But infinitely many solutions is the case when variables are more than the left ones . So there is only unique solution

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